Quantum knot mosaics and the growth constant

TL;DR

This paper investigates the enumeration of quantum knot mosaics, establishing the existence of a growth constant and providing bounds for its value, thus advancing the understanding of quantum knot systems.

Contribution

It proves the existence of the knot mosaic growth constant and derives explicit bounds for its value, addressing an open question in quantum knot enumeration.

Findings

Existence of the knot mosaic constant 

Bounds for : between 4 and approximately 4.303

Provides a mathematical foundation for quantum knot enumeration

Abstract

Lomonaco and Kauffman introduced a knot mosaic system to give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot $n$--mosaic is an $n \times n$ array of 11 mosaic tiles representing a knot or a link diagram by adjoining properly that is called suitably connected. The total number of knot $n$--mosaics is denoted by $D_n$ which is known to grow in a quadratic exponential rate. In this paper, we show the existence of the knot mosaic constant $\delta = \lim_{n \rightarrow \infty} D_n^{\ \frac{1}{n^2}}$ and prove that $$4 \leq \delta \leq \frac{5+ \sqrt{13}}{2} \ (\approx 4.303).$$